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Mirrors > Home > MPE Home > Th. List > funbrfv2b | Structured version Visualization version GIF version |
Description: Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.) |
Ref | Expression |
---|---|
funbrfv2b | ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 6565 | . . . 4 ⊢ (Fun 𝐹 → Rel 𝐹) | |
2 | releldm 5943 | . . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝐵) → 𝐴 ∈ dom 𝐹) | |
3 | 2 | ex 413 | . . . 4 ⊢ (Rel 𝐹 → (𝐴𝐹𝐵 → 𝐴 ∈ dom 𝐹)) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → 𝐴 ∈ dom 𝐹)) |
5 | 4 | pm4.71rd 563 | . 2 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ 𝐴𝐹𝐵))) |
6 | funbrfvb 6946 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) | |
7 | 6 | pm5.32da 579 | . 2 ⊢ (Fun 𝐹 → ((𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵) ↔ (𝐴 ∈ dom 𝐹 ∧ 𝐴𝐹𝐵))) |
8 | 5, 7 | bitr4d 281 | 1 ⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5148 dom cdm 5676 Rel wrel 5681 Fun wfun 6537 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: brtpos2 8219 mpocurryd 8256 xpcomco 9064 fseqenlem2 10022 fpwwe2 10640 joinfval 18328 joinfval2 18329 meetfval 18342 meetfval2 18343 tayl0 25881 ofpreima 31928 funcnvmpt 31930 curf 36552 uncf 36553 curunc 36556 fperdvper 44714 |
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